Influence of Polynomial Chaos expansion order on an uncertain asymmetric rotor system response

Abstract

A stochastic harmonic balance method with a recursive procedure is developed to evaluate the steady-state response of a rotor system with uncertain stiffness and asymmetric coupling that involves time-dependent terms. The Polynomial Chaos (PC) expansion is proposed to evaluate the mean and the standard deviation of the responses and the harmonic amplitudes of orders 1–4 involving 288 degrees-of-freedom. To determine the coefficients of the expansion requires to solve a large system of equations. To achieve this and to avoid numerical problems related to the size of the system, the first contribution of the study consists of a recursive evaluation of the Polynomial Chaos coefficients to be able to estimate the stochastic response for a high order of the Polynomial Chaos expansion.From the methodology implemented, the steady-state responses and n× harmonic components (for n=1, 2, 3 and 4 in the present study) of the asymmetric rotor with uncertainty are evaluated for several PC orders. Then, the second main contribution focuses on a clarification and analysis of the use of Polynomial Chaos expansion around the critical speeds. First, it is observed that the convergence is slow: the response obtained with 30 PCs is twice the one obtained with 200 PCs. Second, it is noted that the parity of the PC order has a strong influence on the response level: whereas two responses obtained with two consecutive even PC orders (or two consecutive odd PC orders) are almost the same for a given rotation speed, the ratio of the responses evaluated with two consecutive PC orders (one even order and one odd order) may be large (e.g. oscillations between two consecutive PC orders greater than 10 are noticeable if the PC order is about 30).